Non-Newtonian fluids and function spaces

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1. Jdayil B. Abu,- Brunn P. O., Effects of nonuniform electric field on slit flow of an electrorheological fluid, J. Rheol. 39 (1995), 1327–1341. (1995) 
2. Jdayil B. Abu,- Brunn P. O., Effects of electrode morphology on the slit flow of an electrorheological fluid, J. Non-New. Fluid Mech. 63 (1996), 45–61. (1996) 
3. Jdayil B. Abu,- Brunn P. O., Study of the flow behaviour of electrorheological fluids at shear- and flow- mode, Chem. Eng. and Proc. 36 (1997), 281–289. (1997) 
4. Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. Zbl 0093.10401, MR 23 #A2610. (1959) Zbl0093.10401MR0125307
5. Bogovskii M. E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), 1037–1040. English transl. in Soviet Math. Dokl. 20 (1979), 1094–1098. Zbl 0499.35022, MR 82b:35135. (1979) MR0553920
6. Bogovskii M. E., Solution of some problemsof vector analysis related to the operators $div$ and $grad$, Trudy Sem. S. L. Soboleva, no. 1, Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk (1980), 5–40. (1980) MR0631691
7. Calderón A. P., Zygmund A., On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. Zbl 0047.10201, MR 14,637f. (1952) MR0052553
8. Cianchi A., Symmetrization in anisotropic elliptic problems, Preprint, 2006. Zbl1219.35028MR2334829
9. Uribe D. Cruz,- Fiorenza A., Martell J. M., Pérez C., The boundedeness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2004), no. 1, 239–264. Zbl 1100.42012, MR 2006m:42029. 
10. Uribe D. Cruz,- Fiorenza A., Neugebauer C. J., The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238. Zbl 1037.42023, MR 2004c:42039. MR1976842
11. Diening L., Maximal function on generalized Lebesgue spaces $L^{p(·)}$, Math. Inequal. Appl. 7 (2004), 245–253. Zbl 1071.42014, MR 2005k:42048. MR2057643
12. Diening L., Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(·)}$ and $W^{k,p(·)}$, Math. Nachr. 268 (2004), 31–43. Zbl 1065.46024, MR 2005d:46071. MR2054530
13. Diening L., Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657–700. MR 2006e:46032. MR2166733
14. Diening L., Ebmeyer C., Růžička M., Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure, SIAM J. Numer. Anal. 45 (2007), no. 2, 457–472. MR2300281. Zbl1140.65060MR2300281
15. Diening L., Ettwein F., Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math. (2006), accepted. Zbl1188.35069MR2418205
16. Diening L., Kreuzer C., Linear convergence of an adaptive finite element method for the $p$-Laplacian equation, SIAM J. Numer. Anal. (2007), submitted. Zbl1168.65060MR2383205
17. Diening L., Málek J., Steinhauer M., On Lipschitz truncations of sobolev functions (with variable exponent) and their selected applications, ESAIM, Control, Optim. Calc. Var. (2006), submitted. Zbl1143.35037MR2394508
18. Diening L., Růžička M., Error estimates for interpolation operators in Orlicz-Sobolev spaces and quasi norms, Num. Math. (2007), DOI 10.1007/s00211-007-0079-9. 
19. Diening L., Růžička M., Calderón–Zygmund operators on generalized Lebesgue spaces $L^{p(·)}$ and problems related to fluid dynamics, J. Reine Angew. Math. 563 (2003), 197–220. Zbl 1072.76071, MR 2005g:42054. MR2009242
20. Diening L., Růžička M., Integral operators on the halfspace in generalized Lebesgue spaces $L^{p(·)}$, Part I, J. Math. Anal. Appl. 298 (2004), 559–571. Zbl pre02107284, MR 2005k:47098a. Zbl1128.47044MR2086975
21. Diening L., Růžička M., Integral operators on the halfspace in generalized Lebesgue spaces $L^{p(·)}$, Part II, J. Math. Anal. Appl. 298 (2004), 572–588. Zbl pre02107285, MR 2005k:47098b. Zbl1128.47044MR2086976
22. Donaldson T., Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations 10 (1971), 507–528. Zbl 0218.35028, MR 45 #7524. (1971) Zbl0218.35028MR0298472
23. Eckart W., Theoretische Untersuchungen von elektrorheologischen Flüssigkeiten bei homogenen und inhomogenen elektrischen Feldern, Aachen: Shaker Verlag.Darmstadt: TU Darmstadt, Fachbereich Mathematik, 2000. Zbl 0958.76003. Zbl0958.76003
24. Ettwein F., Růžička M., Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Comput. Appl. Math. 53 (2007), 595–604. Zbl1122.76092MR2323712
25. Frehse J., Málek J., Steinhauer M., An existence result for fluids with shear dependent viscosity-steady flows, Nonlinear Anal., Theory Methods Appl. 30 (1997), no. 5, 3041–3049. Zbl 0902.35089, MR 99g:76006. (1997) MR1602949
26. Frehse J., Málek J., Steinhauer M., On existence results for fluids with shear dependent viscosity – unsteady flows, Partial differential equations: theory and numerical solution. Proceedings of the ICM’98 satellite conference, Praha, 1998 (W. Jäger, J. Nečas, O. John, K. Najzar and J. Stará, eds.) 121–129, Chapman & Hall/CRC Res. Notes Math. 406, Boca Raton, FL, 2000. (1998) MR1713880
27. Frehse J., Málek J., Steinhauer M., On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (2003), no. 5, 1064–1083 (electronic). Zbl 1050.35080, MR 2005c:76007. Zbl1050.35080MR2001659
28. Giusti E., Metodi diretti nel calcolo delle variazioni, Unione Matematica Italiana, Bologna, 1994. Zbl 0942.49002, MR 2000f:49001. (1994) Zbl0942.49002MR1707291
29. Gossez J. P., Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205. Zbl 0239.35045, MR 49 #7598. (190) MR0342854
30. Halsey T. C., Martin J. E., Adolf D., Rheology of electrorheological fluids, Phys. Rev. Letters 68 (1992), 1519–1522. (1992) 
31. Huber A., Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit $p(·)$-Struktur, Diplomarbeit, Universität Freiburg, 2005. 
32. Hudzik H., The problems of separability, duality, reflexivity and of comparison for generalized orlicz-sobolev spaces $W_m^k\left(\omega \right)$, Comment. Math. Prace Mat. 21 (1979), 315–324. Zbl 0429.46017, MR 81k:46031. (1979) MR0577521
33. Kokilashvili V., Krbec M., Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. Zbl 0751.46021, 93g:42013. (1991) Zbl0751.46021MR1156767
34. Kováčik O., Rákosník J., On spaces $L^{p\left(x\right)}$ and $W^{k,p\left(x\right)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. Zbl 0784.46029, MR 92m:46047. (1991) 
35. skii M. A. Krasnosel,’ Rutickii, Ja. B., Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. Zbl 0095.09103, MR 23 #A4016. (1961) MR0126722
36. Málek J., Nečas J., Rokyta M., Růžička M., Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computations, vol. 13, Chapman & Hall, London, 1996. Zbl 0851.35002, MR 97g:35002. (1996) Zbl0851.35002MR1409366
37. Marcellini P., Papi G., Nonlinear elliptic systems with general growth, J. Differential Equations 221 (2006), no. 2, 412–443. Zbl pre05012954, 2007a:35024. MR2196484
38. Milani A., Picard R., Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems, Partial differential equations and calculus of variations, Lecture Notes Math. 1357, 317–340, Springer, Berlin, 1988, LNM 1357. Zbl 0684.35084, MR 90b:35218. (1988) MR0976242
39. Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. Zbl 0236.26016, MR 45 #2461. (1972) Zbl0236.26016MR0293384
40. Musielak J., Orlicz W., On modular spaces, Studia Math. 18 (1959), 49–65. Zbl 0086.08901, MR 21 #298. (1959) Zbl0099.09202MR0101487
41. Musielak J., Orlicz spaces and modular spaces, Lecture Notes Math. 1034. Springer-Verlag, Berlin, 1983. Zbl 0557.46020, MR 85m:46028. (1983) Zbl0557.46020MR0724434
42. Nakano H., Modulared semi-ordered linear spaces, Tokyo Math. Book Series, Vol. 1. Maruzen Co. Ltd., Tokyo, 1950. Zbl 0041.23401, MR 12,420a. (1950) Zbl0041.23401MR0038565
43. Nekvinda A., Hardy-Littlewood maximal operator on $L^{p\left(x\right)}\left(ℝ\right)$, Math. Inequal. Appl. 7 (2004), no. 2, 255–265. Zbl 1059.42016, MR 2005f:42045. Zbl1059.42016MR2057644
44. Orlicz W., Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211. Zbl 0003.25203. (1931) Zbl0003.25203
45. Parthasarathy M., Klingenberg D. J., Electrorheology: Mechanism and models, Materials, Sciences and Engineering R 17,2 (1996), 57–103. (1996) 
46. Pick L., Růžička M., An example of a space $L^{p\left(x\right)}$ on which the hardy-littlewood maximal operator is not bounded, Expo. Math. 19 (2001), 369–371. Zbl 1003.42013, MR 2002m:42016. MR1876258
47. Rajagopal K. R., Růžička M., On the modeling of electrorheological materials, Mech. Research Comm. 23 (1996), 401–407. Zbl 0890.76007. (1996) Zbl0890.76007
48. Rajagopal K. R., Růžička M., Mathematical modeling of electrorheological materials, Cont. Mech. Thermodynamics 13 (2001), 59–78. Zbl 0971.76100. Zbl0971.76100
49. Růžička M., A note on steady flow of fluids with shear dependent viscosity, Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996). Nonlinear Anal., Theory Methods Appl. 30 (1997), no. 5, 3029–3039. Zbl 0906.35076, MR 99g:76005. (1996) MR1602945
50. Růžička M., Flow of shear dependent electrorheological fluids: unsteady space periodic case, Applied nonlinear analysis (A. Sequeira, ed.), Kluwer/Plenum, New York, 1999, pp. 485–504. Zbl 0954.35138, MR 2001h:76115. (1999) Zbl0954.35138MR1727468
51. Růžička M., Electrorheological fluids: Modeling and mathematical theory, Lecture Notes in Math. 1748, Springer-Verlag, Berlin, 2000. MR 2002a:76004. Zbl0968.76531MR1810360
52. Růžička M., Modeling, mathematical and numerical analysis of electrorheological fluids, Appl. Math. 49 (2004), no. 6, 565–609. Zbl 1099.35103, MR 2005i:35223. Zbl1099.35103MR2099981
53. Samko S. G., Density $C_0^{\infty }\left({ℝ}^n\right)$ in the generalized Sobolev spaces $W^{m,p\left(x\right)}\left({ℝ}^n\right)$, Dokl. Akad. Nauk 369 (1999), no. 4, 451–454. Zbl 1052.46028, MR2001a:46036. (1999) MR1748959
54. Samko S. G., Convolution and potential type operators in $L^{p\left(x\right)}\left({ℝ}^n\right)$, Integral Transform. Spec. Funct. 7 (1998), no. 3-4, 261–284. Zbl 1023.31009, MR 2001d:47076. (1998) MR1775832
55. Sharapudinov I. I., The basis property of the Haar system in the space ${ℒ}^{p\left(t\right)}\left([0,1]\right)$ and the principle of localization in the mean, Mat. Sb. (N.S.) 130(172) (1986), no. 2, 275–283, 286. English transl. in Math. USSR, Sb. 58 (1987), 279–287. Zbl 0639.42026, MR 88b:42034. (1986) MR0854976
56. Sharapudinov I. I., On the uniform boundedness in $L^p(p=p\left(x\right))$ of some families of convolution operators, Mat. Zametki 59 (1996), no. 2, 291–302, 320. English transl. in Math. Notes 59 (1996), no. 1-2, 205–212. Zbl 0873.47023, MR 97c:47035. (1996) MR1391844
57. Talenti G., Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., IV. Ser. 120 (1979), 160–184. Zbl 0419.35041, MR 81i:35068. (1979) Zbl0419.35041MR0551065
58. Wolf J., Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity, J. Math. Fluid Mech. (2006), no. 1, 104–138. MR2305828. MR2305828
59. Zhikov V. V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877. English transl. in Math. USSR Izv. 29 (1987), no. 1, 33–66. Zbl 0599.49031, MR 88a:49026. (1986) Zbl0599.49031MR0864171
60. Zhikov V. V., Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn. 33 (1997), no. 1, 107–114, 143. English transl. in Differential Equations 33 (1997), no. 1, 108–115. Zbl 0911.35089, MR 99b:35170. (1997) MR1607245
61. Zhikov V. V., On some variational problems, Russian J. Math. Phys. 5 (1997), no. 1, 105–116 (1998). Zbl 0917.49006, MR 98k:49004. (1997) Zbl0917.49006MR1486765

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